Three-dimensional topological insulators and topological crystalline insulators represent new quantum states of matter, which are predicted to have insulating bulk states and spin-momentum-locked gapless surface states. Experimentally, it has proven difficult to achieve the high bulk resistivity that would allow surface states to dominate the transport properties over a substantial temperature range. Here we report a series of indium-doped Pb1-xSnxTe compounds that manifest huge bulk resistivities together with evidence consistent with the topological character of the surface states for x ≳ 0.35, based on thickness-dependent transport studies and magnetoresistance measurements. For these bulk-insulating materials, the surface states determine the resistivity for temperatures beyond 20 K.